Let $Y$ be a continuous random variable in $\mathcal{L}^p(\mathbb{R}, \mathcal{B})$ for every $p\in[1,\infty)$ and let $f(x,t)$ be a continuous function, $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, s.t. $\frac{\partial f}{\partial t}(x,t)$ is continuous and, for some $c>0$, $$\left \lvert \frac{\partial f}{\partial t}(x,t) \right\rvert < c \left \lvert x-t \right\rvert^2 ~.$$ Is there any condition or theorem that allows interchanging the derivative and the integral as follows $$ \frac{\partial }{\partial t} \, \mathbb{E} \left[ f(Y,t) \right] \overset{?}{=} \mathbb{E} \left[ \frac{\partial }{\partial t} \, f(Y,t) \right] ~~ ? $$ The well-known dominated convergence theorem cannot be exploited because of the absence of a uniform bound on the derivative. I thought about Vitali's convergence theorem but, again, the uniform integrability hypothesis seems not to be satisfiable.
Thank you in advance.